The downshift paradox - Forced-damped propagation equation

4.2 Forced-damped propagation equation

4.2.3 The downshift paradox

6|a|²∂a

∂t˜ +2a∂|a|²

∂t˜ +2iaH ∂|a|²

∂t˜

| {z }

MNLSE correction

−4k₀³

ω₀νa−20ik³₀ ω₀²ν∂a

∂t˜

| {z }

Viscosity

+k₀

ω₀Γa+4ik₀ ω²₀Γ∂a

∂t˜

| {z }

Wind

(4.2.3)

Comparing to the Dysthe equation eq. (4.2.2), in addition to the leading-order forcing and damping terms, a higher-order forcing term 4i^k⁰

ω²₀Γ^∂a_∂_t_˜ and a higher-order damping term

−20i^k

ω₀²ν^∂a_∂˜_t appear. These terms are of the same order as the Dysthe correction terms to the NLSE, and contain an odd-order derivative. This is very interesting because, as discussed in section 2.4, an odd derivative has an asymmetric action on the spectrum.

k k

Amplification by wind Damping by viscosity

Figure 4.7: Influence on the spectrum of the leading order and higher-order wind terms (left) and viscosity terms (right)

Combining the leading-order and higher-order viscosity terms, all frequencies are damped, but the lower frequencies(k < k₀) are less damped than the higher frequencies, leading to a downshift of the mean. The combined effect of the leading-order and higher-order wind forcing terms is that all frequencies are amplified, but the higher frequencies are amplified stronger than the lower ones, resulting in a relative upshift, as illustrated in fig. 4.7. This figure shows the symmetry of the effect of wind and viscosity.

4.2.3 The downshift paradox

The photos in fig. 2.11 in chapter 2 showed the disintegration of a plane wave in a wave tank.

Figure 4.8 shows the time series measurement of a similar experiment, also launching a slightly perturbed plane wave down a wave tank. Here, after the seeming disintegration, the original wave is actually recovered.

However, the dashed lines show that the period is longer at the end of the tank (x = 30 ft) than at the beginning (x = 5 ft): the frequency has downshifted. The downshift is a heavily researched topic in hydrodynamics. Indeed, if the central frequency is altered, it means a fundamental property has changed. From a more practical point of view, the velocity of the swell is related to its frequency via the dispersion relationship. As we have seen in chapter 3, waves with longer wavelengths travel faster. Therefore, this can play a role in forecasting or

Figure 4.8: Adapted from Yuen and Lake [39]. Evolution of an initially uniform weakly nonlinear wave-train. Distance displayed in feet. The dashed lines show that the wavelength at the beginning of the tank (5 ft) is shorter than at the end of the tank (30ft).

ship routes. Imagine, for example, that a ship needs to leave the harbor before big waves come in, a frequency downshift means that these waves would arrive sooner.

Wind action on the ocean has also been associated with a frequency downshift both in experiment, and simulations [75–79]. This appears as a paradox since in section 4.2.2 we have shown that the viscous damping is associated with a downshift of the spectral mean, as theorized and corroborated with experiments in [80, 81]. The effect of wind forcing has an opposite sign and thus leads to an up-shift of the mean.

It turns out, when looking into each source that associates frequency downshift with wind, that a form of dissipation is at play. This dissipation occurs either in the form of anadverse wind, that effectively damps the waves, or by wave breaking. Therefore, we can lift this paradox by realizing that wave-breaking is an indirect consequence of wind forcing. Wind forcing increases the wave steepness up to the wave-breaking threshold. In turn, wave breaking, like viscous dissipation, leads to a downshift [82–84].

Of course, in an experimental setting, the upshift with respect to the central frequency at the beginning of the tank will be elusive to observe as waves will likely break before it can take place. Rather, it may be observed as counteracting the downshift by viscosity, where the dissipative downshift is lessened with increasing wind speed.

A second source of confusion is whether reference is made to the peak of the spectrum, or the mean of the spectrum. From a mathematical point of view the spectral mean is much more robust. However, experiments often look at the spectral peak. The simulations in the following paper [2] show that in the FPU recurrence cycle, the spectral mean and peak do not behave similarly: the peak downshifts while the mean upshifts, before returning back to their original value. We shall see in section 4.4 ([5] Fig 8 (c,d)) that when the initial condition is a Gaussian spectrum, there is no recurrence cycle, but a similar opposite behaviour of the spectral peak and mean occurs.

The following paper [2], compares the theoretical propagation model 4.2.3 to wave tank experiments at different wind speeds. The initial condition for the wave maker is an Akhmediev breather (fig. 1.9b). In essence, this is a plane wave with sideband perturbations of which we can accurately predict the focal point. This feature is very practical for experiments.

To our knowledge, this is the first time spectra of a model are directly compared to measured spectra. An experimental agreement to NLSE simulations is not possible as it only contains a leading order forcing and damping term, and is spectrally symmetric. Therefore, while the spectrum can change width and amplitude, any asymmetries between k < k₀ and k > k₀ observed in the measurements cannot be replicated. In addition, as can be seen from fig. 2.15, an NLSE model does not predict the correct timescale of the dynamics (i.e. the distance it takes for the sidebands to reach their maximal amplitude). However, with our forced-damped Dysthe equation, we do find the correct timescale and asymmetry, and thus obtain a qualitative agreement with the wind forced spectrum. Combining our model with previous experimental results, we are able to resolve the downshift paradox.

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Dans le document Nonlinear propagation of light pulses and water waves under the influence of damping and forcing (Page 95-111)